---
title: "Impulse response and transfer function significance"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:impulse-response-transfer-significance"
source: https://www.bctengineer.com/q/semester_6/communication_systems/impulse-response-transfer-significance
site: BCT Engineering
---

# Impulse response and transfer function significance

Communication Systems · Chapter 1: Introduction

- Asked in: 2073 Bhadra (CS I), 4+2 marks
- Video: [Impulse Response & Transfer Function of a System](https://www.youtube.com/watch?v=plor6ywAXXg) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/impulse-response-transfer-significance
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

What do you understand by impulse response and transfer functions of a system? Explain its significance.

## Short answer

The impulse response h(t) is the output for a unit impulse; the transfer function H(f)=Y(f)/X(f) is its Fourier transform, giving magnitude and phase response. Both completely characterize an LTI system, enabling output via convolution (time) or multiplication (frequency), and are essential for filter design, stability, and distortion analysis.

## Answer

#### Impulse response
The **impulse response** $h(t)$ of an LTI system is the output produced when the input is a unit impulse $\delta(t)$, with the system initially at rest:

$$
\delta(t) \;\to\; \boxed{\text{LTI system}} \;\to\; h(t)
$$

Because any input can be written as a sum of shifted, scaled impulses, $x(t) = \int x(\lambda)\delta(t-\lambda)d\lambda$, linearity and time invariance give the output as a **convolution**:

$$
y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\lambda)\,h(t-\lambda)\,d\lambda
$$

#### Transfer function
The **transfer function** $H(f)$ is the Fourier transform of the impulse response, and equals the ratio of output to input spectra:

$$
H(f) = \int_{-\infty}^{\infty} h(t)e^{-j2\pi ft}\,dt = \frac{Y(f)}{X(f)} = |H(f)|\,e^{j\theta(f)}
$$

$|H(f)|$ is the amplitude (magnitude) response and $\theta(f)$ the phase response. For an input $e^{j2\pi f_0t}$ the output is $H(f_0)e^{j2\pi f_0t}$.

Example: an RC low-pass filter has $h(t) = \frac{1}{RC}e^{-t/RC}u(t)$ and $H(f) = \frac{1}{1 + j2\pi fRC}$.

#### Significance
- Either $h(t)$ or $H(f)$ **completely characterizes** an LTI system; knowing one, the output for any input can be found.
- Convolution in time becomes **multiplication** in frequency, $Y(f) = H(f)X(f)$, which simplifies analysis of filters, channels and cascaded systems ($H = H_1H_2$).
- $|H(f)|$ and $\theta(f)$ show **bandwidth** and whether the system causes amplitude or phase distortion (distortionless needs $|H| =$ constant, $\theta$ linear).
- $h(t)$ tells whether a system is **causal** ($h(t) = 0$ for $t<0$) and **stable** ($\int|h(t)|dt < \infty$).
- Used to design filters, equalizers and matched filters, and to find output PSD: $S_y(f) = |H(f)|^2S_x(f)$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The transfer function H(f) is:
   A. Inverse of x(t)
   B. Fourier transform of h(t), equal to Y(f)/X(f)
   C. Ratio of x(t) to y(t)
   D. Derivative of h(t)
   Answer: B. H(f) = F{h(t)} = Y/X. (Key point: Transfer function)

2. Convolution in time becomes what in frequency?
   A. Convolution
   B. Division
   C. Addition
   D. Multiplication, Y(f) = H(f)X(f)
   Answer: D. This simplifies analysis; cascaded systems give H = H1 H2. (Key point: Convolution to multiplication)

3. A system is causal and stable when:
   A. h(t) is even and periodic
   B. H(f) is real
   C. h(t) is infinite
   D. h(t)=0 for t<0 and integral |h| dt is finite
   Answer: D. These conditions are read from h(t). (Key point: Causality and stability)

## Related in Communication Systems

- [Impulse response, transfer function and distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adistortionless-transfer-function) — Communication Systems, chapter 1
- [Impulse response usefulness; impulse response of LPF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-usefulness-lpf) — Communication Systems, chapter 1
- [Optimum detector; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-impulse-response) — Communication Systems, chapter 9, asked 3×
- [Optimum detector impulse response in additive white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response) — Communication Systems, chapter 9, asked 2×
- [Impulse response of matched filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-awgn) — Communication Systems, chapter 9, asked 3×
- [Matched filter impulse response is reverse delayed input](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-reverse-delayed) — Communication Systems, chapter 9, asked 2×

## Related in other subjects

- [Unit impulse function and its significance](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aunit-impulse-significance) — Data Communication, chapter 2
- [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2×
- [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5
- [Excitation system with stabilizing transformer: transfer function](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-stabilizing-transformer-model) — Power Plant Equipment, chapter 3, asked 13×